Application of non-local transformations for numerical integration of singularly perturbed boundary-value problems with a small parameter

Singularly perturbed boundary-value problems for second-order ODEs of the form εy_xx ^′′=F(x,y,y_x ^′) with ε→0 are considered. We present a new method of numerical integration of such problems, based on introducing a new non-local independent variable ξ, which is related to the original variables x andy by the equation ξ_x ^′=g(x,y,y_x ^′,ξ). With a suitable choice of the regularizing function g, this method leads to more appropriate problems that allow the application of standard numerical methods with fixed stepsize of ξ (in the whole range of variation of the independent variable x, including both the boundary-layer region and the outer region). It is shown that methods based on piecewise-uniform grids are a particular (degenerate) case of the method of non-local transformations with a piecewise-smooth regularizing function of special form. A number of linear and non-linear test problems with a small parameter (including convective heat and mass transfer type problems) that have exact or asymptotic solutions (both monotonic and non-monotonic), expressed in elementary functions, are presented. Comparison of numerical, exact, and asymptotic solutions showed the high efficiency of the method of non-local transformations for solving singularly perturbed problems with boundary layers. In addition to non-local transformations, examples of the use of point (local) transformations for numerical integration of singularly perturbed boundary-value problems are also given.